this entry is about the notion of genus in algebraic topology/cohomology. For classification of surfaces see instead the (related) entry genus of a surface, genus of a curve. There is also genus of a lattice.
from a cobordism ring for cobordisms with specified structure; typical choices being orientation or stable complex structure. Often the rationalization of such a morphism is meant, see below at Properties – Rationalization.
To emphasize that this is indeed a ring homomorphism and hence in particular respects the multiplicative structure, a genus is sometimes (especially in older literature) synonymously called a multiplicative genus.
respectively. Written this way it is immediate that genera arise naturally as the value on homotopy groups (the “decategorification” or “de-homotopification”) of homomorphisms of E-∞ ring spectra from an actual universal Thom spectrum to some E-∞ ring with coefficient ring
This in turn induces multiplicative morphisms of the cohomology theories represented by these spectra (the domain being hence cobordism cohomology theory), and these multiplicative maps are the “families version” of the given genus/index (Hopkins 94, section 3).
Such homomorphisms in turn arise naturally from universal orientations in generalized E-cohomology. Namely such an orientation is a homotopy of the form
hence a map
At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For some structure, the Thom spectrum is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus can be thought of as assigning to any target space the corresponding sigma-model.
This is for instance the case at least over the point for the A-hat genus , which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus , which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).
is detrmined already by its rationalization
whose generators are identified with the cobordism classes of the manifolds which are the complex projective spaces, as indicated.
Given a (rational) genus one defines (we follow (Hopkins 94))
its logarithm to be the formal power series over given by
its characteristic series (or Hirzebruch series) to be the formal power series over
where is the inverse of the logarithm;
Now rationally, i.e. for , there is a canonical such orientation, given by the composite
Thus, given any orientation , its rationalization may be compared to . Since these rational orientations are equivalently trivializations of maps to , their difference is a class with coefficients in , hence over any space the difference is a class in .
Specifically consider the delooping of the circle group. For this the cohomology ring is the power series ring in a single variable (the universal first Chern class ). Under the canonical inclusion both the above orientations and pull back, so that we have a difference
This is the Hirzebruch series of (Ando-Hopkins-Rezk 10, def. 3.10).
If denotes the formal group law classified via then
This means that is the function of Chern classes (i.e. Pontryagin classes and Euler classes ) obtained by rewriting the polynomial in the (the “Chern roots”) as a polynomial in elementary symmetric polynomials and then substituting for each of these by .
(see also e.g. ManifoldAtlas – Genera – 4.1 Construction).
The Todd genus us the genus with logarithm
The signature genus;
The characteristic series of the -genus is
An elliptic genus is one whose logarithm is given by
for constants with non-degenerate values and .
For degenerate choices this reproduces the signature genus and the A-hat genus above, see at elliptic genus for more. For non-degenrate values one may regard and as values of modular forms of the same name and hence regard all elliptic genera together as one single genus with coefficients in . This “universal” elliptic genus us the Witten genus.
The Witten genus
where are the Eisenstein series (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10, prop. 10.9). (Notice that the constant term in is proportional to the th Bernoulli number, so that indeed the exponential expression matches that for the A-hat genus above.)
On manifolds with rational string structure it takes values in (the -expansion of) modular forms for , meaning that setting then as a function of the parameter taking values in the upper half plane the Witten genus satisfies
|partition function in -dimensional QFT||supercharge||index in cohomology theory||genus||logarithmic coefficients of Hirzebruch series|
|0||push-forward in ordinary cohomology: integration of differential forms||orientation|
|1||spinning particle||Dirac operator||KO-theory index||A-hat genus||Bernoulli numbers||Atiyah-Bott-Shapiro orientation|
|endpoint of 2d Poisson-Chern-Simons theory string||Spin^c Dirac operator twisted by prequantum line bundle||space of quantum states of boundary phase space/Poisson manifold||Todd genus||Bernoulli numbers||Atiyah-Bott-Shapiro orientation|
|endpoint of type II superstring||Spin^c Dirac operator twisted by Chan-Paton gauge field||D-brane charge||Todd genus||Bernoulli numbers||Atiyah-Bott-Shapiro orientation|
|2||type II superstring||Dirac-Ramond operator||superstring partition function in NS-R sector||Ochanine elliptic genus||SO orientation of elliptic cohomology|
|heterotic superstring||Dirac-Ramond operator||superstring partition function||Witten genus||Eisenstein series||string orientation of tmf|
|self-dual string||M5-brane charge|
|3||w4-orientation of EO(2)-theory|
An review of the history is at the beginning of (Hirzebruch-Kreck 09)
The theory of multiplicative sequences and characteristic series of genera is due to
Friedrich Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete, 9, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
Manifold Atlas, Formal group laws and genera
Wikipedia, Genus of a multiplicative series